A Dirichlet Generating Function for the Coefficients of Euler's Pentagonal Number Theorem

Abstract

We establish an integral representation for the Dirichlet generating function of the coefficients of Euler's pentagonal number theorem. The Bromwich-type integral enables analytic continuation to the entire complex plane, filling a gap in the literature and providing a new framework for studying the sequence's analytic structure. Furthermore, we derive the asymptotic behavior as the variable tends to negative infinity, and give integral representations for the Euler function $\phi(q)$ and the Dedekind eta function $\eta(\tau)$. Moreover, we obtain an explicit formula for the Dirichlet generating function at each positive integer, expressed as a finite sum.